International Journal of Advanced Science and Technology

Vol. 54, May, 2013

Multiple Cracks Assessment using Natural Frequency Measurement

Rajarshi Shahu College of Engineering, Tathwade, Pune 411033,

Maharashtra, India2Shri Guru Gobind Singhji Institute of Engg. & Technology, Nanded 431606,Maharashtra, India1prasadbaviskar@gmail.com, 2vbtungikar@gmail.comAbstractThis paper addresses the method of multiple cracks detection in moving parts or beams bymonitoring the natural frequency and prediction of crack location and depth using ArtificialNeural Networks (ANN). Determination of crack properties like depth and location is vital inthe fault diagnosis of rotating machine equipments. For the theoretical analysis, FiniteElement Method (FEM) is used wherein the natural frequency of beam is calculated whereasthe experimentation is performed using Fast Fourier Transform (FFT) analyzer. Inexperimentation, simply supported beam with single crack and cantilever beam with twocracks are considered. The experimental results are validated with the results of FEM(ANSYSTM) software. This formulation can be extended for various boundary conditions aswell as varying cross sectional areas. The database obtained by FEM is used for prediction ofcrack location and depth using Artificial Neural Network (ANN). To investigate the validity ofthe proposed method, some predictions by ANN are compared with the results given by FEM.It is found that the method is capable of predicting the crack location and depth for single aswell as two cracks. This work may be useful for improving online conditioning andmonitoring of machine components and integrity assessment of the structures.Keywords: ANN, Crack, FEM, FFT, Modal Analysis, Natural Frequency

1. IntroductionThe presence of crack in structure changes its dynamic characteristics. The change ischaracterized by change in modal parameters like modal frequencies, modal value andmode shapes associated with each modal frequency. It also alters the structuralparameters like mass, damping matrix, stiffness matrix and flexibility matrix ofstructure. The vibration technique utilizes one or more of these parameters for crackdetection [1-2]. The frequency reduction in cracked beam is not due to removal of massfrom beam, indeed the reduction in mass would increase natural frequency. Butreduction in natural frequency is observed due to removal of material which carriessignificant stresses when defect is a narrow crack or notch [3]. It reduces the stiffnessof structure and natural frequency [4-7]. Due to presence of crack there is localinfluence which results from reduction and second moment of area of cross sectionwhere it is located [8-9]. The system becomes non linear due to crack [10]. Thisreduction is equivalent to lowering the local bending stiffness of beam and therefore itbehaves as two beams connected by means of torsion spring [11-12]. Finite ElementAnalysis is powerful tool which gives the reasonably accurate results for complicated

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on line working assemblies in dynamic analysis [13-14]. Non destructive error detectionsuggests that the variation in monitored signatures is indication of error and it can belocated [15]. Beam forming of lamb waves can also be used for structural healthmonitoring and the results are also promising [16]. To incorporate the non linearity, thecrack is simulated by an equivalent linear spring for longitudinal vibration and thetorsion spring for transverse vibrations connecting the two segments of beam [17]. Theequivalent stiffness may be computed from the crack strain energy function [18]. Theexpression for the spring stiffness representing a crack depth ratio is presented [19]. Assuch correct numerical formula is not available hence the use of 2D element in FiniteElement Analysis is equally valuable. Till now the effect of crack along the width isconsidered. It is observed that the crack along the length does not affect the naturalfrequency up to the considerable mark [20]. This method can be implemented forassessment of multiple cracks also [21-23]. The present study is based on observationof changes in natural frequency. In theoretical analysis, the crack is simulated by aspring connecting the two segments of the beam in the work carried out. For thetheoretical analysis, Finite Element Method (FEM) is used wherein the naturalfrequency of beam is calculated by modal analysis using ANSYSTM whereas forexperimentation purpose, Fast Fourier Transform (FFT) analyzer is used. Inexperimentation, simply supported beam with single crack and cantilever beam withtwo cracks are considered.

2. Analysis of Reduction in Natural Frequencies

A theoretical model based on the receptance technique is presented for analysis. It can betreated as one-dimensional analysis. The crack divides the beam in two sections havingreceptances and respectively. If Kx is the stiffness of the bar, then the natural frequenciesof the cracked bar satisfy the following equation + +

=0

(2.1)

Decrease in Kx is indication of increase in damage. For a bar with uniform cross section arelationship between the crack stiffness, crack location and natural frequency is given by,/ = 1/[ + (1 )]

(2.2)

= 2 /1

(2.3)

Where E, A, and I are the modulus of elasticity, cross-sectional area and length of thebeam respectively. = 1/x is the frequency parameter where is natural frequency of anaxial vibrations. For torsion springs the relationship is in terms of ratios of two determinants.

Where is frequency parameter while 1 and 2 are obtained from the characteristicequation of the system [1] Minimum three modes are required for an efficient prediction. Forthe cantilever beam with multiple cracks (2 cracks), five modes are extracted. Consider EulerBernoulli beam. To derive the differential equation of motion for the bending vibration ofbeam, consider an element of beam of length dx where V and M are shear and bendingmoments. P(x) represents the loading per unit length of beam. Summing the forces in Ydirection,

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Vol. 54, May, 2013

( + ) () = 0, + = ()

(2.4)

By assuming moments about any point on right face of elements in limiting case becomes

(2.5)

The equation (2.4) state that rate of change of shear along the length of the beam is equalto the loading per unit length. Equation (2.5) state that rate of change of the moment along thebeam is equal to the shear. From equation (2.4) and equation (2.5)2 Mx2

= p(x)

(2.6)

Substituting bending moment =

2

= p(x)

2 2

in equation (2.6),(2.7)

For the beam having transverse vibrations, the load per unit length of the beam is theinertia force i.e., mass and acceleration, where M = mass of beam per unit length henceequation (2.7) becomes,22 y2

=M 2 2t 2

4 4

2 2

=0

(2.8)

The equation (2.8) is called governing equation of motion of Euler Bernoulli. Thegeneral solution of the equation (2.8) is obtained by method of separation of variable Y (x,t) =Y(x), T(t), Substituting y = Y.T in equation (2.8),

+ 4 =0 4

1 4 1 (

=) 2 4

The left hand side of the above equation is function of T while the right hand side isfunction of Y alone. It is possible if each side of this equation is equal to negative constantsay - 2 where is a real number.241 = 1 = 2 2 4

1 4

= 2 4

44 4

= 0

4 = 0

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Where4 = equation is,

, from theory of linear differential equation, the solution for above

() = a1 sin x + a2 cos x + a3 sinh x + a4 cosh x

(2.9)

This equation represents harmonic motion of the beam. Where a1, a2, a3 and a4 areconstants and can be found by substituting this solution in the boundary condition. Hence, weget different values of y for the range of x = 0 to 1 for each modes and mode shapes are foundout.

3. Determination of Crack Location

The equation (2.8) of the motion of Euler-Bernoulli does not satisfy near the crack due toabrupt change in the cross-section. The beam can be treated as two uniform beams connectedby a torsional spring at the crack location. The equation (2.8) is then valid for each segmentof the beam separately. This kind of modeling for the cracked beam has the advantage ofusing the exact solution throughout the beam except for a narrow region near the crack wherethe true stress-strain field is approximated by spring. For two beam segments, we get set ofequations from equation (2.9)1() = a1 sin x + a2 cos x + a3 sinh x + a4 cosh x

2() = a5 sin x + a6 cos x + a7 sinh x + a8 cosh x

(2.10 a)(2.10 b)

Where the origin of x for both segments is at the support and 4= A 2/EI. Thecoefficients a1 can be found by substituting this solution in boundary conditions. In case ofstepped beams or shafts, four constants for each step get added. The boundary conditions forsimply supported beam are as follows. For the free vibrations of the beam, Y1A= Y2C = 0 andY1A = Y2C = 0. The continuity conditions at the crack position the displacement, momentsand shear forces are Y1B = Y2B, Y1B = Y2B,Y1B = Y 2B, with the non-dimensional cracksection flexibility denoted by 0, the angular displacement between the two beam segmentscan be related to the moment at this section by, Y2B + L Y"2B = Y'1B. Substituting equation(2.5) in above boundary conditions, a set of eight homogeneous linear algebraic equations forthe eight unknown coefficients is formed [24].11coshesinhecoshe00(sine/)+coshe

+ [sinh(cos cosh) + sin(cose coshe)] = 0

{(cos cose) + sinh(cosh coshe)]}

For uncracked beam with equivalent flexibility, the nominal values of 0 in above equationbecome zero. Substituting 0 = 0 and = n. By definition of the non-dimensional frequencyparameter, 2 / = f/f. Substituting in above equation and rewriting for the ith mode,

f2i = [coseci cosi]fi

For the first natural mode i = 1, above equation yields

f1fi

= (cos2e + 1)

(2.11)

And for the second mode 2 = 2 , Above equation yields,

f2f2

= (cos2e 1)

(2.12)

Dividing equation (2.11) by equation (2.12),

(1 cos 2e)2 1

=(1 + cos 2e)12

Solving above equation for crack location e,

1

= cos 1

21/]21

[1

(2.13)

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Where fn = fn fn, fn and fn are the natural frequencies of uncracked and cracked beam.This relation suggests that the ratio of the relative vibrations of two modes depends solely onthe location of the crack and is independent on crack geometry or beam properties.

4. Determination of Crack Size

Consider a beam with a discrete crack. Considering the characteristic equations, thefrequency change ratio fn/fn and the dimensionless stiffness K is given as,

fn is difference between uncracked and cracked beam = fn fn, fn and fn are the naturalfrequencies of uncracked and cracked beam. The gn(x) function for a simply supported beamcan be evaluated as,1

() = {

4 [( 2 )]

(3.3)

From elementary beam theory, for simply supported beam the mode shape is n = sin (nx). The relationship between the changes in eigen frequencies and the crack location andstiffness of crack based on equation (3.1), (3.2) and (3.3) can be expressed as, = 2 ()/K1 L

(3.4)

The spring stiffness Kr decreases in the vicinity of the cracked section of a beam havingwidth b, height h and crack depth a. From the crack strain energy function, = [(5.346)()]

Using equation (3.6) crack depth ratio (a/h) can be found out if the natural frequency ofbeam is known.

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5. Experimental AnalysisThe instruments used for experimental analysis i.e., measurement of natural frequenciesare Fast Fourier Transform (FFT) analyzer, accelerometer, impact hammer and relatedaccessories. The FFT analyzer used is 4 channel B&K make with measuring range 10-200dB, amplitude stability + 0.1 dB, impedance 10 G , frequency limit 1 Hz to 20 KHz. RTPROTM software, compatible with the FFT analyzer is used. The piezoelectric, miniature typeunidirectional accelerometer is used to capture the frequency response functions. Theaccelerometer is mounted on the beam using mounting clips. The accelerometer is mountednear the crack to capture the correct signal. The impact hammer is used to excite the beamwhose frequency response function has to be captured. For every test, the location of impactof impact hammer is kept constant. Impact hammer has the range of excitation 1-4000 Hz.The beam is tapped gently with the impact hammer. The experiments are performed on mildsteel beams with simply supported boundary conditions having single crack and cantileverboundary conditions with two cracks of different depths at different locations. The propertiesof mild steel are, Youngs modulus (E) 2.0 e11 N/m2, density () 7950 N/m3 and Poissonsratio 0.3. Specimen beams under consideration have rectangular cross section area. Forsimply supported beam the cross sectional area is 0.025 x 0.010 m, L = 0.3m and forcantilever beam the cross sectional area is 0.05 x 0.01m, L = 0.5 m. The geometry of beams isas shown in Figure 1. Crack depth is represented in terms of (a/h) ratio where a = depth ofcrack and h = height of beam and crack location is represented in terms of (e) where e is ratioof location of crack at distance L1 or L2 from the support to the length of the beam L. Theexperimental setup is as shown in Figure 2. The aim of experimental analysis is to verify thepractical applicability of the theoretical method developed. For the beam with single ordouble cracks, transverse and open cracks are considered. Two cracks are parallel to eachother as two cracks of same depth with different orientation do not have any effect on naturalfrequency values [26]. Initially, the natural frequency of uncracked beam is found out.Hairline crack is generated to simulate the actual crack in the working components.Thereafter, the severity i.e., depth of crack is increased. The change in natural frequency dueto the crack is monitored. Table 1 shows the natural frequencies of simply supported beamwith single crack.

6. Finite Element Analysis

Finite Element Analysis is performed using ANSYSTM. The model of beam is generated andused for Finite Element Analysis. The modal analysis is used to determine the naturalfrequencies and mode shapes of a structure. The element used in Finite Element Analysis isPLANE 82: 2-D8-Node Structural Solid. PLANE82 is a higher order version of the twodimensional, four-node element (PLANE42). It provides more accurate results for mixed(quadrilateral-triangular) automatic meshes and can tolerate irregular shapes without loss ofaccuracy [27]. The properties of the material are as mentioned Section 5. Table 1 shows thenatural frequencies of simply supported beam with single crack and Table 2 shows the naturalfrequencies of cantilever beam with two cracks of different depths at different locationsdetermined using ANSYSTM. The crack location and crack depth of first crack with referenceto left support is fixed and the other crack is varied [21].Table 2. Natural Frequencies of Cantilever Beam with Two Cracks by FEMCrack location & size (mm)

International Journal of Advanced Science and Technology

Vol. 54, May, 2013

7. Prediction of Crack Properties by Artificial Neural Networks (ANN)

The inverse problem can be converted into forward technique using tools of ArtificialIntelligence like Genetic algorithm, Fuzzy Logics, Artificial Neural Network (ANN). Thesetechniques can be used for prediction of life of components or even optimization to minimizethe errors in frequencies determined by numerical simulation and experimental measurement.Genetic algorithms are stochastic search algorithms which are based on the mechanics ofnature selection and natural genetics. These are designed to search large, non-linear, discreteand poorly understood search space where expert knowledge is difficult to model andtraditional optimization techniques may not give accurate results. In the genetic algorithm,this error is used to evaluate the fitness of each individual in the population. Geneticalgorithms have been frequently accepted as optimization methods in various fields and havealso been proved as an excellent in solving complicated optimization problem. Thus, GeneticAlgorithm can be used to solve inverse problem for the crack detection in a shaft [28]. TheArtificial Neural Networks (ANN) in a wide sense belongs to the class of evolutionarycomputing algorithms that try to simulate natural evolution of information handling [29]. Thepresent paper checks the applicability of this tool to predict the crack location and depthdepending upon the input. The input to the ANN is the natural frequency of three or morenumber of modes and output is crack location and crack depth. In case of single crack theoutput will be prediction of crack location and crack depth i.e., two parameters whereas fortwo cracks the output will be prediction of four parameters i.e., two predictions for crackdepths and two predictions for crack locations. Amongst the available data, 90% data is usedto train the network in ANN whereas 10% of the data is used for validation. The backpropagation algorithm is used [30]. The network is trained using the data obtained by FEMi.e., Table 1 for simply supported beam with single crack and Table 2 for cantilever beamwith two cracks. Thereafter, the network predicts the location and depth of crack. Thenetwork can predict the crack location and depth for any intermediate input values of naturalfrequencies. The network decides the predominant input parameter on its own. The iterationsare conducted till the average training error and average validating error is minimized [31].For simply supported beam with single crack, single layer serves the purpose whereas forcantilever beam with two cracks, three layers give close predictions. For cantilever beam withtwo cracks, three layers are chosen as average training error and average prediction error isminimum in case of three layers. Less error in both is indication of precise prediction ofoutput. During the routine assessment of the health of component or online conditioning andmonitoring, if decrease in natural frequency is observed, these frequency values can be givenas input in the form of new query to the network. The network predicts the properties ofcrack. Any number of queries can be run.

8. Results and Discussions

The crack of known severity is generated at known location in mild steel beam. In case ofsimply supported beam single crack is generated whereas in case of cantilever beam, twocracks are generated. The changes in natural frequencies for the uncracked and cracked beamsare measured. The predicted values are determined by theoretical and experimental technique.Table 1 shows the natural frequency values extracted for simply supported beam with singlecrack determined by using FEM and experimentation. Non dimensional frequency ratio is alsocalculated using these values. By the inverse method i.e., by using equation 2.13 and 3.6 theresults for crack location (e), crack size (a/h) are computed. Table 3 shows the comparison forcrack location (e), crack size (a/h) between actual and determined values for simply supportedbeam with single crack. It compares between theoretical and experimental frequency ratio

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(f1/ f1) with respect to crack size (a/h). The theoretical results i.e., results obtained by FEMare compared with the experimental results and Figure 3 shows graphical comparisonbetween the theoretical (FEM) and experimental non dimensional frequency ratio. It isobserved that experimental results have some deviation from the results obtained by FEM asmodel of structure generated by Finite Element Analysis differs from actual structure. Hencethe response of structure in practice differs. The results are close to the actual for finding thecrack locations. These results approach to the actual results found by FEM as compared to theexperimental findings. The variation in the results obtained is in the range of 0.2 to 15%. Thevariation might be the effect of structure prone vibrations. The results of crack depth findingsare close to the actual depth for large (a/h) ratio as compared to small (a/h) ratio. It is observedbecause for small (a/h) ratio, the reduction in the stiffness of beam is less as compared to large(a/h) ratio. Due to the high stiffness, the vibrations are damped and natural frequency does notreduce. The readings obtained are used as database for Neural Networks. ANN can predict thecrack location and crack depth by adding new query to ANN grid. The predicted cracklocation and depth by ANN are verified by using FEM. Table 4 shows the comparison ofcrack properties like crack depth and crack location predicted by ANN with the resultsobtained by FEM with the actual. The results are in close agreement. Similar procedure isextended for two cracks at different locations with varying severity in case of cantileverbeam. The accuracy in prediction of crack properties is more for single crack than two cracks.

Figure 3. Comparison of Theoretical and Practical Results

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Table 3. Comparison of Crack Location and Size Determined for Single Crack(Experimental method and FEA)Actual (mm)a/he0.10.20.10.40.10.60.10.80.20.20.20.40.20.60.20.80.30.20.30.40.30.60.30.80.40.20.40.40.40.60.40.8

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9. ConclusionThis work attempts to establish a systematic method of prediction of crack characteristicsfrom measurement of natural frequencies using ANN. From the numerical and experimentalstudy, following conclusions can be drawni.ii.iii.

iv.

v.

vi.

vii.

viii.

Variation in natural frequencies of first two to five modes is observed as they arepredominant in crack properties.The results of Finite Element Analysis and experimental analysis are compared and theyare in good agreement.For the same severity of crack, the frequency reduction is more for location of crackaway from the support because of the stiffness of the structure; vibrations getsuppressed near the supports.The error in prediction of crack location by theoretical analysis is in the range of 3% to15% where as in case of experimental analysis; it is in the range of 5% to 20%. Thevariation in the experimental results is due to structure prone vibrations and vibrationsgetting transmitted through foundation.The database obtained is used to as input to train the Neural Network. Appropriatelytrained Network can predict crack characteristics like depth and location by giving thenatural frequency as input.The predictions of crack location and depth by ANN are verified with the results ofFEM. The results are in good agreement with error of 1% to 5% for single crackwhereas up to 15% for multiple cracks.In the present study, the beams under consideration have uniform cross section but thismethod can be extended to components with varying cross section, different geometryand any boundary condition.The proposed method can be extended for fault diagnosis in beams, shafts or rotatingmachine element.

[2] M. Karthikeyan, R. Tiwari and S. Talukdar, Development of a technique to locate and quantify a crack in abeam based on modal parameters, Journal of Vibration and Acoustics, vol. 129, (2007) June, pp. 395-401.

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AuthorsV. B. Tungikar has completed M.Tech. from I.I.T., Kharagpur andPh.D. from S.R.T.M. University, Nanded. He has 25 years of Teachingand 04 years of Industrial experience. Presently, he is working asProfessor in Production Engineering Department at Shri Guru GobindSinghji Institute of Engineering and Technology, Nanded. His areas ofinterest include FEA in the field of thermal and structural analysis. Hiswork on thermo elastic analysis of composites, wear of metal matrixcomposites is published in the Journals of national and internationalrepute. He has completed 03 research projects funded by CentralGovernment of India under the schemes of All India Council forTechnical Education (AICTE), New Delhi.P. R. Baviskar has completed M.E. Mechanical Engineering fromPune University, Pune and is Ph.D. research scholar in S.R.T.M.University, Nanded. He has 15 years of teaching experience. He has beenworking as Assistant Professor in Mechanical Engineering Department atRajarshi Shahu College of Engineering, Pune. His areas of interest areMachine Design and Mechanical Vibrations. He has published researchpapers in International Journals and International Conferences onanalysis of crack in beams / shafts using vibration technique and FEM.